What are the "simplest" examples of countable groups that are not known to be sofic?

$\begingroup$ I think Thompson's groups are not known to be sofic. See Pestov's survey: google.com/… $\endgroup$ – Ian Agol Feb 16 '14 at 5:56
The simplest candidate I know of is Higman's group
$\langle a,b,c,d\mid a^b=a^2, b^c=b^2, c^d=c^2, d^a=d^2\rangle$
(where, as usual, $a^b$ means $b^{1}ab$). Terry Tao wrote a nice blog post about it here.
Residually finite and amenable groups are both known to be sofic. As Tao explains, Higman's group has no finite quotients (hence is 'highly nonresidually finite') and a nonabelian free subgroup (hence is certainly nonamenable).

$\begingroup$ Thank you very much! Are there any nonresidually finite nonamenable groups that are known to be sofic? $\endgroup$ – Vladimir Feb 10 '14 at 18:03

4$\begingroup$ Soficity is preserved under direct products, so just take a direct product of nonamenable and non residually finite sofic groups, such as a free group and a BaumslagSolitar group like $\langle x,y \mid y^{1}x^2y=x^3 \rangle$. $\endgroup$ – Derek Holt Feb 10 '14 at 18:21

1$\begingroup$ A more recent reference appears here: arxiv.org/abs/1512.02135 $\endgroup$ – Ian Agol Aug 19 '16 at 4:51